Integrals I: Feynman's Trick · Superchallenge problem

Suppose $x$ is a real number with $x + \dfrac1x$ an integer. Induction shows $x^n + \dfrac{1}{x^n}$ is an integer for every $n \ge 1$. If $x + \dfrac1x = 3$, compute $x^5 + \dfrac{1}{x^5}$.